Dense Subset Sum in Multi-Dimension

2026-07-11

AI Summary

This work addresses the challenges of insufficient integer-point coverage and low algorithmic efficiency in the high-dimensional dense subset sum problem. By integrating tools from additive combinatorics, the theory of high-dimensional generalized arithmetic progressions, and geometric characterizations of zonotopes, the authors prove that for any fixed dimension $d$, whenever the set size satisfies $n \gg \sqrt{\Phi}$, its subset sums cover all integer points within a prescribed zonotope. The study extends the optimal one-dimensional density threshold to higher dimensions, substantially reducing the density required for full coverage. Furthermore, it introduces a near-linear time decision algorithm with complexity $\tilde{O}(n)$, achieving simultaneous advances in both theoretical bounds and computational efficiency.

Abstract

We study the additive structure of dense subset sum in multi-dimension, and use the structure to develop efficient algorithms for the dense subset sum problem. More precisely, given a set $A$ of $n$ vectors in the $d$-dimensional hyperrectangle $[N_1]\times [N_2]\times\cdots\times [N_d]$, we study the structure of $\mathcal{S}(A)$, which is the set of all subset sums of $A$. We focus on the dense regime of the problem where $n \gg \sqrtΦ$ and $Φ= N_1 \times \cdots \times N_d$. We show that for any constant $d\geq 1$, if $n \gg \sqrtΦ$, then $\mathcal{S}(A)$ contains a long generalized progression in multi-dimension. If we further have that no non-trivial lattice can contain the majority of $A$, then $\mathcal{S}(A)$ contains all the integer points in the zonotope $\{x_1\vec{a}_1 + \cdots + x_n\vec{a}_n: o(1)\leq x_j \leq 1-o(1), x_j \in \mathbb{R}\}$. Compared to the previous results for $d \geq 2$, our result significantly reduces the density threshold and enlarges the region inside which all the integer points belong to $\mathcal{S}(A)$. Also, it matches the bound for the 1-dimensional case. Using our combinatorics result, we also develop an $\tilde{O}(n)$-time algorithm for the dense subset sum problem in multi-dimension.

Problem

• dense subset sum
• multi-dimension
• subset sums
• additive combinatorics
• zonotope

Innovation

• dense subset sum
• multi-dimensional
• generalized arithmetic progression
• zonotope
• additive combinatorics